# Questions tagged [forcing]

Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).

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### Proving $\|x=y\|\cdot \|\phi(x)\|\le\|\phi(y)\|$ in Boolean valued models

This question relates to the Boolean algebra approach to forcing. Fix a complete Boolean algebra $B$. I'm writing $\|\sigma\|$ for the Boolean value of $\sigma$, where $\sigma$ is a sentence of the ...
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### Role of Negation in Tarski Truth and Cohen Forcing Definitions

As I am new to Forcing, I would appreciate any help on whether the following is anywhere near being correct : Given a Structure M, Enderton, 2001, "A Mathematical Introduction to Logic" defines truth ...
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### Are the $\mathsf{HOD}$s preserved by weakly homogeneous forcings?

We saw the following theorem in class: Let $M$ be a transitive model of $\mathsf{ZFC}$, let $\Bbb P\in M$ be a weakly homogeneous partially ordered set, let $G$ be $\Bbb P$-generic over $M$ and let ...
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### Forcing with restricted condition: it's definition [closed]

What does it mean to apply the symbol $\Vdash$ to a condition $q$ restricted to $\xi$: $q\upharpoonright \xi\Vdash\ldots$ as used on the page 5 above lemma 2.4 here; is this $\upharpoonright$ there ...
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### Elementary example of Baire spaces whose product is not Baire

It is known that there are Baire spaces $X$ and $Y$ whose product is not Baire, the simplest construction I know is due to Cohen and goes as follow: Let $S$ be a stationary subset of $\omega_1$, then ...
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### Cohen Set Theory and the Continuum Hypothesis p44 Partial Truth Formulae

In Cohen, Set Theory and the Continuum Hypothesis, page 44 the ability to form Partial Truth Formulae is described : "We leave as an exercise for the reader the proof of the following fact: For each ...
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### In Chow's “beginner's guide to forcing”, why is $\bigcup U$ a function?

I'm reading Timothy Chow's A beginner's guide to forcing (in a quest to finally familiarize myself with "boolean-valued model" forcing), and this passage on page 13 threw me for a loop: ... let $P$ ...
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### Shooting a club is Baire

I'm attempting this problem from Kunen: I'm trying to do it by a direct combinatorial argument. Namely, let $C$ be the set of countable ordinals for which there exist such an $\omega$-chain. I want ...
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